324 ANALYTICAL MECHANICS 249. Logarithmic Decrement. — The logarii hm of the ratio of two consecutive amplitudes is constant and is called the logarithmic decrement of the motion. The amplitudes occur whenever the relation CO* tan (at) = — a is satisfied. Let the first amplitude occur at the instant t = h; then since the period of the tangent is TT, the times of the succeeding amplitudes are given by tan (wf) = tan (wti + HT) , u / / _L 7l7r or by t = Cl + ~~ » CO where n is a positive integer. Hence, denoting the logarithmic decrement by X and the nth amplitude by anj we have X = log —2- (by definition) a» + 2 - ^i - Sin^wfi ± " " "'"" A6""flvl " ...... « v sin [wfe + (n + 2) a(,rf^) ! 6 V w/ = log - r-TTTT 2ir ~~~ — 0) = ~P. (XXV) Therefore if I is known k" may be determined from observations of P and a. * Obtained by setting^- » 0.